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SECULAR   PERTURBATIONS 


ARISING    FROM    THE 


ACTION  OF  SATURN  UPON  MARS 


AN  APPLICATION  OF  THE  METHOD  OF  ARNDT 


A  THESIS 

PRESENTED  TO  THE  FACULTY  OF  PHILOSOPHY  OF  THE 
UNIVERSITY  OF  PENNSYLVANIA 

BY 
SAMUEL  GOODWIN    BARTON 

IN  PARTIAL  FULFILMENT  OF  THE  EEQUIREMENTS  FOR  THE 
DEGREE  OF  DOCTOR  OF  PHILOSOPHY 


PHILADELPHIA 

1906 


SECULAR    PERTURBATIONS 


ARISING    FROM    THE 


ACTION  OF  SATURN  UPON  MARS 


AN   APPLICATION    OF    THE   METHOD    OF  ARNDT 


A  THESIS 

PRESENTED  TO  THE  FACULTY  OF  PHILOSOPHY  OF  THE 
UNIVERSITY  OF  PENNSYLVANIA 

BY 
SAMUEL  GOODWIN   BARTON 

IN  PARTIAL  FULFILMENT  OF  THE  REQUIREMENTS  FOR  THE 
DEGREE  OF  DOCTOR  OF  PHILOSOPHY 


PHILADELPHIA 
1906 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 
LANCASTER,  PA. 


INTRODUCTION. 

The  paper  upon  which  this  thesis  is  based  is  entitled  "  Re- 
cherches  sur  le  calcul  des  forces  perturbatrices  dans  la  theorie 
des  perturbations  seculaires"  by  Dr.  Louis  Arndt.  The  paper 
is  published  as  a  bulletin  of  the  "  Societe  des  Sciences  Naturelles 
de  Neuchatel,"  an  extract  from  tome  XXIV,  1896. 

Aside  from  the  application  of  the  author,  as  a  check  upon  his 
formulae,  mentioned  in  the  paper,  so  far  as  is  known,  no  applica- 
tion of  this  method  has  been  made.  The  purpose  of  the  author  of 
this  thesis  is  to  apply  this  method  of  computation  and  to  compare 
it  with  that  of  G.  W.  Hill  and  thus  see  whether  Arndt's  method 
has  the  advantages  claimed  for  it  by  its  author. 


I. 
EXPRESSIONS  FOR  THE  SECULAR  PERTURBATIONS. 

Let  R,  U,  Z  (each  multiplied  by  the  factor  containing  the  masses, 
i.  e.,  m  I(L -\-  m)  be  the  components  of  the  perturbing  force; 
R  acting  in  the  plane  of  the  orbit  of  the  perturbing  body  and  in 
the  direction  of  its  radius  vector,  r,  positive  if  r  is  increased ;  U  in 
the  plane  of  the  orbit  perpendicular  to  the  radius  vector,  positive 
in  the  direction  of  movement ;  Z  perpendicular  to  the  plane  of  the 
orbit,  positive  northward.  Let  fl  be  the  longitude  of  node,  i  the 
inclination  of  the  orbit  to  a  fixed  ecliptic,  a  the  semi-major  axis, 
e  the  excentricity,  /A  the  mean  daily  motion,  a>  the  angular  distance 
of  perihelion  from  the  ascending  node,  v9  E,  M,  the  true,  eccentric 
and  mean  anomalies  and  u  =  CD  -j-  v. 

In  accordance  with  the  usual  method  of  varying  parameters  we 
have  the  following  equations  for  the  variations  of  the  elements. 

3 


165260 


4 


map  sec<t>  _        . 

sin  »(8fl)  =  —j^  -  -  f  sin  u  Z,     (S»)  =     -,          r  cos 


[sin  «B  +  (cos  v  +  cos 


(So))  =  (8a>,)  —  COS  i(Sfl), 


(I) 


*  +  2  sin'     (&0  +  2  sin° 

The  secular  part  of  these  expressions  will  comprise  those  terms 
which  are  independent  of  the  positions  of  the  bodies  in  their  orbits, 
that  is  of  their  mean  anomalies  M  and  M  '  .  Each  of  the  above 
variations  is  seen  to  be  a  function  of  7?,  £7,  and  Z  which  in  turn 
are  functions  of  M  and  M'  .  They  may  therefore  be  expressed  in 
a  Fourier's  series,  i.  e.,  in  a  series  of  sines  and  cosines  of  multiples 
of  M  and  M  '  .  Now  the  terms  of  this  expansion  which  are  inde- 
pendent of  M  and  M'  give  the  secular  variation.  By  the  known 
properties  of  the  series  the  only  constant  term  with  respect  to  M 
is  given  by  the  integral 


and  the  part  of  this  which  is  constant  with  respect  to  M'  will  be 


Thus  the  secular  part  of  any  variation  (Se)  for  example  is 

TO'"V  C°S  *  •  i  r  r  [sin  vR  +  (cos  v  +  cos  E)  U^lMdM  '. 

1       -|-      m  47T       JQ  JQ 

Since  the  limits  of  integration  are  constant,  sin  v,  cos  v  and 
cos  E,  may  be  treated  as  constants  in  the  integration  with  respect 
to  M'  i  and  this  integral  may  be  written 

m'a*p  cos  <f>  1    £**     .         \    f- 

-  ^—          fsm  v  ~~         RdM 

1  +  m      27rJ0      L          27rJ0 

-f  (cos  v  -f  cos  E)  ~  I**"  UdM']dM. 

"rjr  Jo 


Treating  the  other  variations  in  the  same  manner  we  see  that 
the  integration  with  respect  to  M'  requires  the  three  integrals 


1        /»»«• 

fcf 


The  integration  with  respect  to  Jlf  can  be  obtained  only  by  a 
direct  quadrature.  Since  this  can  be  obtained  more  accurately  in 
terms  of  E  we  transform  M  and  M'  in  our  expressions  into  E 
and  E  '.  Then  since 

M=E-e  sin  E,        dM=  (1-6  cos  E)dE=-dE, 
dM'  =  (l-er  co%E')dE'. 

Our  desired  integrals  which  which  we  shall  now  designate 
7?0,  Z70,  Z0  respectively  are 

~§\rR(\  -  c'  cos  .E")dtf',  ^  Par  £7(1  -  e'  cos 
(2) 

-^  f  ^  r2^!  -  e'  cos  j0')cLtf' 

where  the  factors  a  and  r  have  been  taken  with  the  integrals. 


II. 
PRELIMINARY  CONSTANTS. 

Let  the  orbit  of  the  perturbing  body  m  be  referred  to  that  of 
the  perturbed  body  m.  Let  the  distances  of  the  perihelia  from 
the  intersection  of  the  orbits  be  II  and  II'  and  their  mutual  incli- 
nation J.  Let  K  and  K'  be  the  distances  of  the  intersection 
from  the  nodes.  Solving  the  triangle  thus  formed  we  obtain 
IT,  K'  and  J  in  terms  of  i,  i'  and  (ft'  —  ft)  whence  we  get  II  and 
IT  from  the  relations 

n  =  TT  —  ft  -  K,     IT  =  if  —  ft'  -  K'. 

Let  L'  and  B'  be  the  longitude  and  latitude  of  m  referred  to 
the  orbit  of  m  and  the  intersection.  Let  £  77,  %  be  rectangular 
axes,  origin  at  the  sun,  77  and  f  lying  in  the  plane  of  the  orbit  of 


6 

w,  f  passing  through  the  body  m.     Let  I  be  the  longitude  of  m. 
The  coordinates  of  m  are  then  given  by  the  equations 

I  «  H  -f  v,  f  =  /  cos  #  cos  (Lr  -  Q, 

T/  =  r'  cos  .7?'  sin  (£'  —  Z),     £"  =  /  sin  B. 

Now  from  the  right  triangle  formed  by  the  intersection,  m\  and 
the  foot  of  the  perpendicular  from  m  to  the  orbit  of  m  we  have 

cos  L'  cos  E'  —  cos  (II'  -f  v),     sin  U  cos  .Z?'  =  sin  (II'  +  v')  cos  J] 

sin  5'  =  sin  (IT  +  v)  sin  «7. 
To  simplify  the  formulae,  write 

A  sin  A'  =  —  sin  II'  cos  <7,     vlc  =  Ad  cos  (J/  +  II  +  v), 

4,  =  Ba!  cos  <£'  sin  (^'  +  II  +  v), 
J.  cos  A  =  cos  II',  Be  =  —  JLa'  sin  (J/  +  H  +  v)r 


(3) 


Bt  =  J?a'  cos  <#>'  cos  (J^  +  ft  +  v), 
sin  B'  =  —  sin  II',  (7C  =  a'  sin  II'  sin  «/", 

C7t  =  a'  cos  $'  cos  II'  sin  «7, 


Substituting  these  expressions  in  £'  for  example :  by  expansion 
f '  =  /  cos  II'  cos  (II  -f-  v)  cos  v  +  r  sin  II'  sin  (II  -f  v)  cos  «7cos  v 

—  r  sin  IT  cos  (II  -f  v)  sin  v  -f  r  cos  II'  sin  (II  +  v)  cos  </sin  vV 
but 

r  cos  v  =  a'(cos  ^"  —  e'),     /  sin  -y'  =  a'  cos  $'  sin  ^", 
whence 

•-  =  ^.X008  ^^  ~ e')  +  ^« sin  ^'»   and  similarly> 

;'  =  C  (cos  ^'  -  e')  +  C.  sin  j^', 


7 


III. 

EXPRESSIONS  FOR  JRQ,  £70,  Z0. 
With  axes  as  described  it  is  not  difficult  to  see  that 


A3     '         ~A3'          ~A3' 

where  A  is  the  distance  between  the  two  bodies.     We  see  that 
A2  =  (f  '  —  rf  +  77'  -f  J"  which  from  (4)  and  (3)  becomes 

A2  =  A0  -  2#0  cos  (e  -  E")  +  CQ  coslE" 
in  which  we  have  placed 

«'2  -f  r2  +  2eWLc  =  ^0,  eV2  +  r^4c  =  ^0  cos  e, 

V    /  ^  7-)       .  ^2   /2  x> 

rAs  =  ^0  sin  e,  a  e    =  (70. 

Substituting  the  values  of  7?,  £7,  Z  in  (2)  we  obtain 

"      ^c(cos  ,E"  -  e')  -f  ^  sin  ^  -  r 
==  ~ 


(6) 


J9c(cos  E'  —  e)  -f  Bt  sin  E' 

(1  —  e'  cos  E' 
C2n   „  C  (cos  .£"  —  e')  4.  Q  sin  _£" 

'•-•i^J,    r~  -SF 

(1  -  e'  cos  E'}dE'. 

These  expressions  cannot  be  directly  integrated  because  of  the 
complexity  of  A  in  terms  of  E'.  A  transformation  due  to  GAUSS 
(  Werke  III,  p.  333)  makes  integration  possible  by  changing  the 
variable  E'  to  a  new  variable  T. 


8 


-r  =  - 


aa"  + 


IV. 

GAUSS'S  TRANSFORMATION. 
Let 

E'  =  a  +  a'  sin  T  +  a"  cos  T, 

E'  =  0  +  /3'sin  T+  j3"  cos  T, 
=  7  +  7'  sin  J7  +  7"  cos  J7. 
Now  let  these  new  auxiliaries  be  subjected  to  the  conditions 

r  -  77'  =  o, 

—  77"  =  0, 
"'  -  7'7"  =  0, 

2  rZ  f/2  -t  Q  t  ,y  "  Q"  A 

a  —  a    —  a     =  —  1,         ap  —  ap  —  a  p    =  U, 

?2  —  ft'   —  /3"   =  —  1,  ay  —  a'y  —  a"y"  =  0, 

72  —  y'Z  —  y"Z  =  1,  fty  —  ft'y'  —  ft"y"  =  0. 

We  now  make  these  auxiliaries  such  that  A2  may  take  the  form 

(a)  ^2A2  =  G  -  Gl  sin2  T+  G2  cos2  T. 

Assuming  this  possible  we  shall  see  if  real  values  of  the 
coefficients  can  be  found  such  that  ^V2A2  may  take  the  form 

E'  •  NCOS  e 

i^'- JVsin€+  CJiNunEJ. 

We  now  solve  (7)  for  sin  T  and  cos  T  in  terms  of  E'  and 
observing  that  1  =  —  aN  cos  E'  —  $N  sin  E'  +  yN  and  substi- 
tuting these  values  in  (a),  then  equating  coefficients  with  (b)  and 
writing  the  resulting  equations  in  three  groups  we  have 

I  a  G  •  a  —  a  Gl  •  a  -j-  a"  G2  •  a"  =  C0 
SG-a-pGt-a'  +  P'Gz'a'^Q 
yG'a  —  y'G^a'  +  y'Gz-a'^  BQ  cos  e 

The  last  two  columns  are  the  values  of  similar  equations,  the  a's 
being  replaced  by  /3's  and  7's  respectively. 


(9) 


0 
0 
sin  e 


,0086 

,  sin  e 

A 


9 
Whence 


GOL  ==  —  CQOL  -f  yBQ  cos  e, 
G/3  =  7#0  sin  e, 

Gy  =  —  a^?0  cos  e  —  £J90  sin  e  -f  yAQ, 
GjLi  etc.,     and     —  6r2a",  etc., 

give  the  same  expressions  with  a,  /3,  7,  replaced  by  a',  y8',  7'  and 
a"»  £"»  7  "•  The  coefficients  of  the  a's,  /3's  and  y's  are  of  the  same 
form  in  each  set  of  equations.  The  condition  that  these  equations 
may  be  consistent  is  expressed  by  equating  the  determinant  of  the 
coefficients  of  a,  /3,  7  to  zero.  This  gives  when  expanded 

#[(#  +  Q(#  -  A)  +  Bl  cos2  e]  +  Bl(G  +  <70)  sin2  e  =  0. 

Thus  if  N2A2  is  to  be  of  the  desired  form  (a),  G  must  satisfy 
this  equation,  and  since  the  other  two  groups  give  the  same  equa- 
tion, except  for  accents,  Gl  and  —  G2  must  also  satisfy  this  equation. 

Making  the  variable  X  and  placing 

(10)    P^A.-Ca     Pt-Bl-AtC.     -P3=<70^sin'e 

the  cubic  may  be  written 

(  C)  X3  -  P^2  +  P2X-  P3  =  0 

GGi  and  —  G2  are  then  the  roots  of  this  equation.  The  roots  can 
be  shown  to  be  real,  two  positive  and  one  negative.  We  let  them 
be  in  order  of  descending  magnitude  G,  Gl  and  —  G2. 

Multiply  numerator  and  denominator  of  (6)  by  Nz  and  the  de- 
nominator will  be  of  the  desired  form  (a).  As  shown  by  Gauss 
NdE'  —  dT  and  the  limits  for  T  are  the  same  as  those  for  E  '. 
Because  of  the  limits  terms  containing  sin  T,  cos  T  or  sin  Tcos  T 
vanish  and  our  transformation  gives 

rm  V_R  TT  z__  i  fV  r+  r.  sin*  r+r, 

^'^'^-2^1    M(G-GlStf  r+<? 
In  which  we  place 

(•  T  =/72  +  bay  +  A/87  -  da/3  -  la\ 

(12)        J  r,  =yy2 


=/7"2  +  lay"  +  W°l"  -  da"/3"  -  la"2. 


10 


For  S0. 
/  =  —Ae'  —  r 

For  UQ. 
~^X 

For  Z0. 
-Gf 

=  AJl  +  e2)  +  re' 

^c(l  +  ^/2) 

Cc(l  +  e'2) 

A  =  A 

5. 

c-. 

d  =  ^x 

Be 

Gf 

Z  =  Ae' 

Be' 

Gf 

M  —  ar 

ar 

r* 

It  is  only  in  the  integration  of  (11)  that  Arndt's  method  differs 
essentially  from  that  of  Hill.  Hill's  method  consists  of  integration 
by  means  of  elliptic  integrals.  The  modulus  of  the  elliptic  in- 
tegrals is  &=*(Gl+  G2)/(G+  G2).  The  computation  of  this 
modulus  necessitates  a  solution  of  the  cubic  (  (7)  by  approximations 
for  the  roots  6r,  6^,  —  6r2.  While  the  solution  is  not  difficult  it 
becomes  objectionable  because  of  the  number  of  times  it  must  be 
made.  Arndt's  method  avoids  the  solution  of  this  cubic  by  mak- 
ing the  integration  depend  upon  the  integrals  of  Weierstrass. 


V. 

AKNDT'S  MODIFICATION. 

LetTF=  r, .dr         _=  f?^. 

Jo    y  6^  —  G!  sin  T  -\-  G2  cos  ^Z7     Jo     ^ 5 

then 

dW  fi  dT 


N* 
TdT 


Our  expression  for  V  then  becomes 


11 

Now  let  sin2  T=  t  and  (G1  +  #,)/(#  +  #,)  =  ^  (modulus),  then 
£  =  0  when  r=  0,  £  =  1  when  T=  Tr/2  and 


and 

z> 

£7  \jf 

where 


X1         5TF  rl          dw       rl 

Qdt,    SG=-P\   Qtdt,   j£  =  P     Q(l-t)dt 
U^TI  Jo  v^z  Jo 


4(G 


and 


In  order  to  use  Weierstrass's  function  for  the  integration  we 
make 


where  s  is  a  new  variable  and  m,  77i1?  m2  constants  which  will  be 
disposed  of.     By  differentiation  we  obtain 

tfdt  dt 

^' 


(k2  —  l)dt 
whence 


, l2- 
When  t  =  0,  3  =  (^x  when  ^  =  1,  s  =  —  (r2.     Now  writing 

m  =  !/((?  -  G,)     and     (ff  +  G2)(G  -  ^)(^  +  ^2)  =  (7 


and  ix4(s  —  G)(s  —  G^s  +  G2)  =  V S^  we  have 

dW      G,  +  Gn   r~G*  ds 
^G 

dW 


aw_     G+G2.r-«*  ds 

3G=          '20     JGl     ^       ° 


12 

In  order  that  S  may  be  of  the  Weierstrassian  form  the  second 
term  of  the  cubic  ( (7)  must  be  wanting,  that  is,  each  root  must  be 
diminished  by  Pj/3.  Hence 

£— Vsi 


where  e^  e2,  e3  are  the  roots  of  the  transformed  equation.     We 
then  have 

8W      Gl+G2  r*~_^ds 


,  +  g,  r 

•20      1 


dG  '       '2C      ^    v/^ 

5TF          G+  G2  pa  —  ea 
'X   VI 

5TT     Q-Gr'-^j. 


SO,-  ^    V~8 

r 

•/* 


a^2~     2(7 

and 

_ L+_2(elft>  +  ^) 


7T 


r«  ds 
..  1/ 


We  must  now  obtain  the  coefficients  of  o>  and  77  in  terms  of  the 
elements.     For  this  purpose  let 

A  =  (&l  +  G2)T  +  (G 
@  =  (0,  +  GJGT  +  ( 
then,  since  e,  =  G  -  ^P,,  e2  -  G,  -  JP15  es  =  -  <?2  -  £ 


A  and  ©  must  now  be  expressed  in  terms  of  the  coefficients  rather 
than  in  terms  of  the  roots  of  (  (7).     Let 


and  from  the  symmetric  functions  of  the  roots 

^  GG2-  G,G2  =  P2, 


13 

Substituting  the  values  of  F,  T1<t  T2  given  in  (12)  we  see  that 
the  equation  for  Al  reduces  to  Al  —  —f—  I  which  by  (13)  for 
the  three  cases  gives  respectively  r,  0,  0. 

By  substituting  from  (12)  and  reducing  with  (9)  we  find 

Bl  =fA0  +  bB0  cos  e  +  hBQ  sin  €  -  I  <70, 
whence  from  (13) 


f  =  -  AQ(Aee'+  r)  +  B0  cos  €{eV 
(15) 

B*  —  A9C.e'+  -#0cos  e(7c(l  +  e2)  +  BQO8  sin  e  -  <7c<7/. 


For  5^  it  is  preferable  to  substitute  the  primitive  values,  whence 
we  obtain 


(16)         J?f  =  -  Jra'  cos2  ^  sin2  ,/sin  2(11  +  t>)  -  ^'^c- 

For  reducing  Cl  we  multiply  the  equations  Got2—  0^+  G2a"2=z  CQ 
and  G  +  #!  -  ^2  =  Pl  by  a'2  -f  a"2  -  a2  =  1  and 
GG1—  GG2—  0^2=  P2  respectively  and  add  the  products. 
We  thus  have 


a'G.G,  -  a2GG2  -f  a'GG,  =  P2  -f  Pl  C0  +  a/(72  +  a^  -  «2(72. 

The  last  three  terms  can  be  replaced  by  known  quantities.     Mul- 
tiplying 

Go.  =  —  C0a  +  BQ  cos  67      by     —  Ga  , 

^0  cos  €7  Gjot, 


and  adding  the  products  we  obtain 


~  ^2«2  =  Cl  -  ^2  cos2  e. 


Dividing  by  G&G  =  —  P3  we  get 

a2       a         <t"2.ggina6 


Similarly  we  obtain 


14 


dff      a"/3"       1    ^    , 

-rr  -f  -7T-  =  ^  -#o  sm  e  cos  e, 


«7      a  7       ay 
3y      Py    .   /3"7" 


cos  e 


sin. 


-Cosine 


The  three  terms  in  C^  occur  in  the  form 


Vb  +  ^h      a/3d      -I 
b  h  G*  ~  G      ~  G 

The  other  two  quantities  are  given  by  the  same  equation  with  ap- 
propriate accents.     Substituting  in  the  equation  we  find 

-  PsCi  =  BQ  sin  e(cLS0  cos  e  -  IBQ  sin  €  -  hCQ). 
Substituting  from  (13)  for  each  case  we  find 


Ar'e'ABa'2  cos  <£'  cos  (.4'  -  H), 


To  obtain  A  we  consider  the  identity  (v.  HalpJien  FonGtions 
Uiptiques  ®ch.  VIII). 


3 


-  r   i\    r2 

111 

0    0i  -02 

• 

111 
(?  ^  ""^"2 

= 

111 

/^  /"*  /^ 

tr  Crr  —  Cr2 

Multiply  by  the  equation  —  GG1G2  —  PB  using  a  factor  in  each 
column  of  the  second  determinant.  Subtract  the  last  column 
from  each  of  the  first  two  and  upon  expanding  we  find  it  equal  to 
+  C.  In  the  first  determinant  multiply  the  first  column  by 


15 


_  (  Q^  _j_  £y  and  multiply  the  last  two  columns  by  (  G  +  6r2)  and 
(Gl  —  G)  respectively  and  add  them  to  the  first  column.  We 
thus  find  for  this  determinant  thus  modified : 


-A 


1 

A 
0 
0 

r, 

ff, 

i 

r2 

-«i 
i 

G1+G2 

Our  identity  when  expanded  then  becomes 


-  B^P,  -  9P3)  - 
Now  let 
(18)  P1Pt-9P,-P,     P?-3P2 

We  then  obtain  for  the  three  cases 


•J  -  3Pa). 


(19)      H 


If    we   write    the   transformed   cubic   in    the   usual    manner 
4S3  _  g^s  —  g3  =  0,  we  have  the  following  equations  for  the  roots. 


e2  +  e3  =  0, 

—  4(  0  £,-££,- 


P2 


If  gz  and  g3  are  the  invariants  of  the  cubic  whose  roots  are  6r, 
the  discriminant  is  g\  —  27#*  and  C72,  i.  e.,  the  product  of  the 
squared  differences  of  the  roots  will  be 

(21)  C". 


(22) 

g  is  the  absolute  invariant. 


16 


Again  for 


G 


®  we  consider 

an  identity 

G2T2 

1 

1 

1 

-G. 
1 

- 

1 

G2+6 

1 
^ 

1 
G, 

G 

~          0, 

G, 

.-A,     A 
P,     3        0 


-p.- 

Multiply  this  equation  by  —  GGtG2  =  P3,  using  a  factor  in  each 
column  of  the  second  determinant,  the  negative  sign  being  used  in 
the  last  column.  Next  subtract  the  last  column  from  each  of  the 
first  two  ;  the  second  determinant  when  thus  treated  and  expanded 
will  be  found  equal  to  2\(  Gl  +  6r2).  In  the  first  determinant 
multiply  the  columns  respectively  by  ( G^  +  6r2),  ( G  -f  6^),  and 
(Gl  —  G).  Subtract  the  last  two  columns  from  the  first.  The 
determinant  thus  treated  and  expanded  will  become  ®j(Gl  +  G2). 
Our  identity  then  becomes  2\O  =  A(P1P2—  9P3)  +  C(P^ 
Whence  we  have  for  the  three  components 


(23) 


and  for  the  components  we  have  finally  from  (14) 
2       FA*/  P, 


(24) 


2 


P 


The  elliptic  integrals  w  and  77  are  computed  by  the  method  of 
H.  BRUNS.*     Writing  for  the  moment  ^o>a  =  ea  we  know  that 

{«K  4-  u)  -  ea}(s  -  ea)  =  (ea  -  e^(ea  -  ey). 


*"Ueber  die  Perioden  der  elliptischen  Integrate,"  Math.  Ann.,  Bd.  27. 


17 

Then 
/  = 


(s  - 


Let 


•.  s  = 


5   — 

ds  = 7-7 — 


,         e  e2g3 

ei  H ir—      — , 


When 

5  =  ea,     s'  = 

ds 


5  =  ea,     s  =  <?3,     s  =  cs,     s  =  e2, 


£*  ds         r*  a'da' 

^  +  ->7S"JL^— 


If  now  ^2  and  ^3  are  given  functions  of  f  ,  we  obtain,  by  differ- 
entiating the  equations  for  co  and  77, 

d<»     p  (?;  +  &),,    ^_      f'(y;«  +  y3)  ,. 

df-J.,       2^i      rfs'     d|~   -Je,  ' 
where 

/          d$2  A  '          dff* 

^-?     and     ^3  =  ^. 
But  we  have  the  identities 


,- 6^-63)   (.-«,)• 


18 

and  introducing  the  integral  I 


sds 


whence 

(25)  ~=-Pr,-Qa>,    | 

where 


t 
' 


or  expressed  in  terms  of  the  symmetric  functions  of  the  roots 

-  -  86^;  + 


If  now  we  substitute  for  &>  and  97  in  (25)  respectively 


10 


the  coefficients  of  the  differential  equations  can  be  expressed  by  g 
and  its  derivatives  only  and  we  obtain 


n 

-i)1 

and  now  for  g'd%  we  write  dg. 

Eliminating  H  and  H  respectively  from  these  equations  we  get 

.   d£l       55 


19 


These  equations  are  of   the  form  of   the  differential  equation 
whose  solution  is  the  hypergeometric  series,  i.  e., 

0  =  x(x-  1)        +  [>(a+  fl  +  l)-7]         +  apy 


where  a  =  $,  ft  =  T62  ,  7  =  f  ,  and  a  =  T\,  £  =  —  ^  7  =  J.     One 
of  the  24  solutions  of  this  equation  is 


y 


O)  = 


To  determine  the  constants  (7,  (7'  we  place 
=  1,  0r3  =  1,  then  #2  =  3,  and  e^l,  e2=  e3=  -  },  t?  =  ei~e*  =  Q, 


and  since 


1         C  d6 

CO  =  -  =^:^  -  ^ 

l/6i  —  e3  Jo     VI  —  /fc2  si 


TT 

— 

sin2  <^  1/6 


l  —  e3  1/24 


since  when  g  =  1,  (</  —  !)/#  =  0,  and  F  —  1 


.-.    ^  ==^d  I       »»t>»  •      —  =  £||  ••          . ,     V^    ==^J  I  "t»     -          =  4d  i  «  — , 

1/6  iX!2  1/24  i!/1728 

whence 

(26)  «-j^ 


where  JP,  and  ^  are  the  two  hypergeometric  series 

For  computation  we  first  solve  the  triangle  described  in  part  II, 
for  the  preliminary  constants,  then  apply  the  formulae  in  the  fol- 
lowing order  3,  5,  10,  18,  20,  22,  21,  15,  16,  17,  19,  23,  26,  24. 
The  values  of  Fm  and  F^  required  in  26  may  be  taken  from  the 


20 

table  given  by  Arndt  or  they  may  be  computed  directly.     The 
argument  of  this  table  is  x  =  (g  —  l)/gr. 
We  now  compute  the  following  expressions  : 

HU  —  ZQ  sin  u,     H{  —  ZQ  cos  u,    He  —  7?0  sin  v  -f  £70(cos  v  +  cos  E), 


1  +  sin 


r 

1  +  5(1^ 


We  now  take  the  mean  value  of  each  quantity,  which  we  repre- 
sent by  (fffi),  etc.,  and  obtain  the  variation  of  the  elements  from 
the  equations :  m> 

=  1  +ra' 

sin  £(Sfl)  =  B  sec  <£(^T0),         (Si)  =  B  sec 
(8e)  =  B  cos  <t>(ffe),     e(^wi)  =  -S  cos 

—  cos 


(SJQ  +  2  sin2^^^)  4.  2  sin2  1  (an). 

X  and  TT  are  the  mean  longitude  and  the  longitude  of  perihelion 
measured  from  a  fixed  epoch. 


VI. 

COMPUTATION. 
Saturn  upon  Mars. 

The  elements  taken  from  Dr.  G.  W.  Hill's  "  New  Theory  of 
of  Jupiter  and  Saturn  "  are 

Mars  p.  192.  Saturn  pp.  19,558. 

7r=333°17'51".74  TT'=   90°    6'4r.37 

i=      1    51      2.24  i'=      2    2940.19 

H  =    48    23    54.59  ft'  =  112    20  49  .05 

e  =  0.09326803  e  =  0.05606025 

P  =  689050".784  p'  =  43996"21506 

log  a  =  0.1828971  log  a  =  0.9794956 

m  »  1/3093500  m  =  1/3501.6 
Epoch  1850.0  G.M.T. 


21 


The  preliminary  constants  are  found  to  be : 


n  =  176°17'59".42 
IT  =  293  4  38  .78 

K=  108  35  57  .73 
1C  =  44  41  13  .54 

J=      2    2152  .11 


A  =  0.99927955 
#  =  0.9998659 
^'  =  66°54'17".84 
B'  =  66    56  24  .55 


The  values  of  the  various  quantities  given  by  the  formulae  are 
given  in  the  table  below.  The  residual  in  Innes's  test  equation 
was  found  to  be  —  0.000,000,000,2.  2t  and  22  are  the  sums  of 
the  values  for  the  odd  and  even  points  of  division 


E 

log  r 

V 

log  A. 

log^c 

log  A. 

logJ?g 

0 

0.1403760 

o°  o'  o!6o 

0.6331692* 

0.9298510 

0.9295583* 

0.63221  33n 

30 

0.1463201 

32  47  24.62 

9.9980540 

0.9768010 

0.97634  55n 

0.0001597 

60 

0.1621568 

64  44  46.64 

0.7680496 

0.8760042 

0.8753687* 

0.7679641 

90 

0.1828971 

95  21  5.91 

0.9480287 

0.5421705 

0.5410633n 

0.9477062 

120 

0.2026919 

124  31  47.16 

0.9752134 

0.1081758n 

0.1097082 

0.9747498 

150 

0.2166314 

152  34  23.40 

0.8883581 

0.7460753* 

0.7460180 

0.8877384 

180 

0.2216237 

180  0  0.00 

0.6331692 

0.9298510n 

0.9295583 

0.6322133 

210 

0.2166314 

207  25  36.60 

9.0215124n 

0.9791564n 

0.9787259 

9.0446347* 

240 

0.2026919 

235  28  12.84 

0.6602801n 

0.9223582n 

0.9217848 

0.6603403* 

270 

0.1828971 

264  38  54.09 

0.9069056n 

0.7051686n 

0.7043164 

0.9066458n 

300 

0.1621568 

295  15  13.36 

0.9790254n 

9.4089485* 

9.3984933 

0.9786051n 

330 

0.1463201 

327  12  35.38 

0.9148835n 

0.6835510 

0.6835778n 

0.9142995n 

~sT 

1.0916971 

900  0  0.00 

2, 

1.0916972 

1080  0  0.00 

E 

10g50 

log^0 

p 

\ 

g 

logC" 

0 
30 

1.0710308 
1.1693425 

1.9648845 
1.9689849 

10682.6599 
18227.1670 

8117.2267 
8041.3776 

1.0027367 
1.0056470 

8.3349592 
8.6360598 

60 

1.2416284 

1.9733790 

26314.7192 

7960.4005 

1.0092812 

8.8370927 

90 
120 

1.2868582 
1.3068094 

1.9769311 
1.9787352 

32935.6960 
36424.3060 

7895.1366 

7862.0877 

1.0129818 
1.0155661 

8.9705083 
9.0427814 

150 

1.3022786 

1.9783135 

35792.7330 

7869.9176 

1.0157721 

9.0497000 

180 

1.2725893 

1.9757393 

31049.5750 

7917.3434 

1.0131888 

8.9809495 

210 
240 

1.2158096 
1.1302863 

1.9716558 
1.9671494 

23362.5420 
14847.5260 

7992.6957 
8076.0143 

1.0087905 
1.0043632 

8.8189886 
8.5302017 

270 
300 

1.0231950 
0.9419157 

1.9634684 
1.9616475 

7941.6038 
4595.4802 

8144.1096 
8177.6560 

1.0013537 
1.0002558 

8.0341618 
7.3163722 

330 

0.9692790 

1.9621814 

5654.5662 

8167.4836 

1.0008465 

7.8342202 

1 

11.8215349 
11.8215351 

123914.27 
123914.31 

48110.729 
48110.721 

6.0453918 
6.0453916 

22 


E 

lOgPstf/ 

log-P.Ci* 

B* 

B  u 

Bp 

A*C 

0 

1.9169034 

0.5318556 

—1.71845 

—  .89671773 

0.3721019 

1273741.5 

30 

1.9755785 

0.5328775 

—3.01775 

-1.13441580 

—2.5062115 

2172708.2 

60 

1.9062750 

0.2070736 

-4.62374 

—  .98414281 

—4.7095692 

3211393.9 

90 

1.6134503 

8.6889828n 

—6.08172 

—  .44676768 

—5.6474343 

4168957.1 

120 

1.2216849n 

9.5472262 

—6.93032 

0.29124699 

—5.0684299 

4791819.1 

150 

1.8858736,, 

0.4342181 

—6.88475 

0.95971490 

—3.1277779 

4857869.0 

180 

2.0793986n 

0.6943511 

—5.96030 

1.34015180 

—0.3456203 

4287900.0 

210 

2.1185815n 

0.6960869 

—4.47493 

1.35536890 

2.5324875 

3228683.9 

240 

2.0337615n 

0.4411836 

—2.90359 

1.07265140 

4.7354355 

2023242.3 

270 

1.7767034» 

9.5893663 

—1.66622 

0.62356451 

5.6730943 

1059477.9 

300 

0.4293996« 

8.6103629W 

—1.00508 

0.10695994 

5.0942969 

597208.9 

330 

1.6828108 

0.2039025 

—1.01624 

—  .42731476 

3.1540537 

697610.1 

2, 

23.14148 

0.9301496 

0.0782150 

16185305.7 

23.14161 

0.9301500 

0.0782128 

16185306.2 

E 

0 
30 
60 
90 
120 
150 
180 

A"C- 

AZC 

*$ 

&u 
log  -c- 

^ 

1,000  R0 

1331148.8 
1499656.7 
1257141.7 
633680.8 
-  251362.3 
-1175895.1 
—1859502.8 

—  51269.99 
—100538.88 
—149578.11 
—185230.51 
-190157.71 
—154729.51 
-  89067.22 

8.0562601 
8.1210086 
8.1877509 
8.2373935 
8.2639876 
8.2663024 
8.2448728 

7.5893686 
7.5702475 
7.4530372 
7.1237659 
6.6740144M 
7.3428058« 
7.5509401« 

5.9407458n 
6.8638621n 
7.095934U 
7.1721544n 
7.1355044* 
6.9590236n 
6.3943817n 

1.7097689 
1.7745204 
1.9262896 
2.1375744 
2.3569481 
2.5168682 
2.5606371 

210 
240 

—2068748.7 
—1729849.5 

—  20232.80 
25701.71 

8.2024945 
8.1483792 

7.6367383n 
7.6523844n 

6.6338873 
6.9773713 

2.4720554 
2.2843136 

270 

—  969088.3 

38725.85 

8.1153154 

7.6286630n 

7.0861514 

2.0608640 

300 
330 

I 

—  43468.5 
784500.8 

24077.58 

—  8288.07 

8.2046672 
8.0486528 

6.7557748n 
7.5910007 

7.1006984 
7.7301623 

1.8653206 
1.7413404 

—1295892.6 
—1295893.9 

—430293.74 
—430293.93 

12.70328 
12.70322 

E 

1000  U0 

1000  Z0 

1000  #n 

1000  Ht 

1000  Hu 

1000  He 

-500  HMl 

0 

+.0099129364 

+.01219913 

—.01178899 

+.00313663 

—1.7097689 

+0.01982587 

1.5503021 

30 

+.0070243377 

—.10023809 

+.06747523 

—.07412671 

—1.4844357 

+0.97300477 

1.6311880 

60 

+.0054967025 

-.20386986 

+.03664317 

—.20054972 

—0.8120554 

+1.74728269 

1.8364587 

90 

+.0055926247 

-.27213016 

—.09419268 

—.25530876 

+0.2105525 

+2.12773530 

2.1375744 

120 

+.0032050993 

—.27013335 

—.20519340 

—.17569210 

+1.3414279 

+1.93830949 

2.4668628 

150 

—.0049176511 

—.17936971 

—.17509923 

—.03890714 

+2.2292371 

+1.16793251 

2.7201623 

180 

—.016050410 

—.02369097 

—.02289447 

+.00609141 

+2.5606371 

+0.03210082 

2.7994626 

210 

—.022805479 

+.13539885 

+.10010378 

—.09117062 

+2.2161522 

—1.09867500 

2.6717296 

240 

—.019925518 

+.23709997 

+.07965469 

--.22331936 

+  1.3285740 

—1.86063280 

2.3908407 

270 

—.0090535750 

+.25685432 

—.04260349 

—.25329642 

+0.2103199 

—2.05103630 

2.0608640 

300 

+.0029944062 

+.20818148 

—.13424167 

—.15911853 

—0.8010985 

—1.68427280 

1.7783332 

330 

+.0097886250 

+.11927107 

—.11350343 

-.03664085 

—1.4740911 

—0.92634119 

1.6006880 

2i 

—.014366784 

—.04021460 

—.25782067 

—.74945050 

+1.907705 

+0.19261327 

12.8222601 

*\ 

—.014371118 

—.04021372 

—.25781983 

—.74945177 

+1.907735 

+0.19262010 

12.8222063 

23 


The  resulting  values  are 


(&») 
e(S7T) 
(SZ,) 


sn 


-  ".024687315, 
".0062897022, 
".062294586, 
".93075550, 

".062281786, 

-  ".83828575, 

-  ".0084927392. 


For  comparison  we  give  the  results  of  Newcomb  and  of  Lever- 
rier  and  those  obtained  by  applying  Hill's  method.*  Leverrier's 
values  are  reduced  to  the  value  of  m  here  adopted. 


Newcomb. 


Leverrier. 


Method  of  Hill.     Method  of  Arndt. 


(*) 


(<*) 

(6L) 


+  0.00629 

+  0.00628 

+  0.0062891 

+  0.0062897 

+  0.06226 

+  0.06226 

+  0.0622814 

4-  0.0622817 

—  0.00849 

—  0.00852 

—  0.0084927 

—  0.0084927 

—  0.02468 

—  0.02467 

—  0.0246873 

—  0.0246873 



—  0.838 

—  0.8382821 

—  0.8382857 

CONCLUSION. 


The  computation  apparently  shows  that  this  method  is  some- 
what less  accurate  than  that  of  Hill.  The  cause  lies  in  the  fact 
that  the  formulae  involve  differences  of  almost  equal  quantities. 
This  is  true,  in  this  problem  at  least,  in  the  formulae  for  .Z?f,  C2 
and  UQ.  For  example,  the  formula  for  C2  (21)  contains  the  factor 
(g  —  1)  ;  a  glance  at  the  values  of  g  given  in  the  table  shows  that 
this  factor  becomes  so  small  that  its  logarithm,  and  consequently 
that  of  (72,  becomes  quite  inaccurate  in  its  last  places.  Whether 
these  conditions  are  accidental  to  the  problem  is  not  evident.  In 
spite  of  the  greater  mathematical  elegance  of  the  treatment  by 
Arndt,  it  is  the  opinion  of  the  author  that  the  additional  accuracy 
secured  by  applying  Hill's  method  would  repay  the  extra  com- 
putation. 

*A.  J.,  No,  574. 


